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Hardware/Binary/Hexadecimal_number_system.md
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@ -61,12 +61,41 @@ $$
As applied to `1A5`:
| $16^{2}$ | $16^{1}$ | $16^{0}$ |
| --------------------- | -------------------------- | ------------------- |
| $1\cdot 16^{2} = 256$ | $A (10)\cdot 16^{1} = 160$ | $5\cdot 16^{0} = 5$ |
| $16^{3}$ | $16^{2}$ | $16^{1}$ | $16^{0}$ |
| ---------------------- | --------------------- | -------------------------- | ------------------- |
| $1\cdot 16^{3} = 4096$ | $1\cdot 16^{2} = 256$ | $A (10)\cdot 16^{1} = 160$ | $5\cdot 16^{0} = 5$ |
Another example for `F00F`:
$(15 \cdot 4096 = 61440) + (0 \cdot 256 = 0) + (0 \cdot 16 = 0) + (15 \cdot 1 = 15) = 61455$
## Using hexadecimal to simplify binary numbers
Whilst computers themselves do not use the hexadecimal number system (everything is binary), hexadecimal offers advantages for humans who must work with binary:
1. It is much easier to read a hexadecimal number than long sequences of binary numbers
2. It is easier to quickly convert binary numbers to hexadecimal than to convert binary numbers to decimal
Look at the following equivalences
| Number system | Example 1 | Example 2 |
| --------------- | ------------------- | ------------------- |
| **Binary** | 1111 0000 0000 1111 | 1000 1000 1000 0001 |
| **Hexadecimal** | F00F | 8881 |
| **Decimal** | 61,455 | 34,945 |
It is obvious that a pattern is maintained between the hexadecimal and binary numbers and that this pattern is obscured by the decimal conversion. In the first example the binary half-byte `1111` is matched by the hexadecimal `F00F`.
Mathematically comparing hex `F` and binary `1111`
$$
\textsf{1111} = (2^{3} + 2^{2} + 2^{1} + 2^{0}) \\
= 8 + 4 + 2 + 1
$$
$$
\textsf{F00F} = (15 \cdot 16^{4}) + (15 \cdot 16^{0}) \\
= 8 + 4 + 2 + 1
$$
// TODO: Relation to binary and bytes